Show the following rule for MVDs The attributes are arbitrar

Show the following rule for MVD\'s. The attributes are arbitrary sets X, Y, Z and the other unnamed attributes of the relation in which these dependencies hold. The Difference Rule. If X, Y, and Z are sets of attributes, X rightarrow rightarrow Y, and X rightarrow rightarrow rightarrow rightarrow Z, then X rightarrow rightarrow rightarrow rightarrow (Y-Z).

Solution

Assume that W be the set of attributes not in X, Y, or Z, V be the set of attributes that Y, Z have in common, Y1 be the set of attributes of Y not in V and Z1 be the set of attributes of Z not in V. Consider the two tuples xy1vz1w and xy1\'v\'z1\'w\'. Because X®®Y, swap the y\'s so tuples xy1\'v\'z1w and xy1vz1\'w\' are in R. Because X®®Z, obtain the pair xy1\'v\'z1w and xy1vz1w swap the z\'s to get xy1\'vz1w and xy1v\'z1w. Because X®®Z, take the pair xy1vz1\'w\' and xy1\'v\'z1\'w\' and swap the z\'s to get xy1v\'z1\'w\' and xy1\'vz1\'w\'. In final, started with tuples xy1vz1w and xy1\'v\'z1\'w\' and showed that xy1\'vz1w, xy1v\'z1\'w\' should also be in the relation. That is accurately the statement of the MVD X®®(Y – Z).